cuspidal cubic curve

This cubic is one of my favourite curves. It has equationy2=x3. If you draw a picture
you'll
see a sharp point at the origin where there isn't a well-defined
tangent. This is a singular point of the curve.

| .
|
| /
| .
| .
|--------------------
| .
| .
| \
|
| .

In fact this singularity is what stops the humble cusp from being a mighty elliptic curve. There is a bijectivesmoothing parametrisation of the curve given
by t --> (t2,t3).

If you intersect the complex points of the cusp
with a small sphere|x|2 + |y|2=e2, for a small positive real
number e then with some calculation you can show that
this intersection is topologically the same as the trefoilknot.